The Aerodynamics of Sails 



non-uniform Kb A 



? . (37) 



Typical values of Kb/4Ug are on the order of 0.03. 



The condition for minimum induced drag at fixed lift is constant induced 

 angle. As shown in Table 2, the circulation distribution needed to result in 

 constant induced angle xmder normal conditions has only the first four terms of 

 its Fourier series representation in Eq. (9) significantly different from zero. 

 For a fixed value of the lift, the strength of the second term needed for minimum 

 induced drag is the quantity most affected by the presence of the image plane and 

 the wind gradient. Under normal conditions these effects oppose each other 

 (Table 2). The fact that the wind strength increases with height reduces A from 

 its value in a uniform wind for minimum induced drag. The presence of the 

 image plane increases A from its value on an unbounded airfoil for minimum 

 induced drag. For most cases the image-plane effect slightly outweighs the 

 velocity gradient effect, and A2 is small and positive for minimum induced drag. 



For a fixed lift, the largest effect on the heeling moment is that due to Aj . 

 Furthermore, the way to alter the load distribution from that giving minimum 

 induced drag, such that the heeling moment is changed the most for the least in- 

 crease in induced drag, is to alter A2 . The above facts coupled with the fact that 

 most sails can support more circulation over their lower portions than over their 

 upper portions, because of differences in local chord length, indicate a general 

 scheme for the design of vertical load distributions. This is to choose Aj to 

 give the desired amount of lift, and Aj to prevent excessive heeling moment and 

 excessive local lift coefficients near the head of the sail. All the other An's 

 should be almost zero. 



THE THEORY OF TWO LIFTING LINES 

 AS APPLIED TO SAILS 



Within the limits of linearized theory, the lift, induced drag, and heeling 

 moment of a system of staggered airfoils are independent of the stagger. This 

 is a consequence of Munk's (1918-1921) equivalence theorem for stagger which 

 states that the total induced drag of a lifting system is unaltered if any of the 

 lifting elements are translated parallel to the free stream direction. This 

 theorem is true because such a translation causes no change in the flow in the 

 Trefftz plane. By the same theorem, airfoils can be contracted to lifting lines 

 for purposes of determining lift, drag, and heeling moment. Therefore, the lift, 

 drag, and heeling moment for a sloop-rigged vessel can be determined by con- 

 tracting the mainsail to the mast and the jib to the jibstay. The problem is then 

 that of a pair of skewed lifting lines. 



The drag of a sailing rig is dominated by the induced drag. Therefore, 

 sloop rigs can be evaluated by determining the lift and induced drag, and the 

 resulting forward force, side force, and heeling moment for the pair of skew 

 lifting lines representing the mainsail and jib, A computer program has been 

 prepared to do this in the presence of an image plane and a linear velocity pro- 

 file. The program has been checked with known analytical results, and forces 

 obtained by the two methods vary by about one percent. 



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